Name
Abstract | ||
|---|---|---|
Finite dimensional local convergence results for self-adaptive proximal point methods and nonlinear functions with multiple minimizers are generalized and extended to a Hilbert space setting. The principle assumption is a local error bound condition which relates the growth in the function to the distance to the set of minimizers. A local convergence result is established for almost exact iterates. Less restrictive acceptance criteria for the proximal iterates are also analyzed. These criteria are expressed in terms of a subdifferential of the proximal function and either a subdifferential of the original function or an iteration difference. If the proximal regularization parameter $\mu({\bf x})$ is sufficiently small and bounded away from zero and $f$ is sufficiently smooth, then there is local linear convergence to the set of minimizers. For a locally convex function, a convergence result similar to that for almost exact iterates is established. For a locally convex solution set and smooth functions, it is shown that if the proximal regularization parameter has the form $\mu({\bf x})=\beta\|f'[{\bf x}]\|^{\eta}$, where $\eta\in(0,2)$, then the convergence is at least superlinear if $\eta\in(0,1)$ and at least quadratic if $\eta\in[1,2)$. |
Year | DOI | Venue |
|---|---|---|
2007 | 10.1137/060666627 | SIAM J. Control and Optimization |
Keywords | Field | DocType |
new class,self-adaptive proximal point method,proximal regularization parameter,asymptotic convergence analysis,convex function,proximal function,proximal point methods,local convergence result,convergence result,local linear convergence,proximal iterates,finite dimensional local convergence,exact iterates | Mathematical optimization,Mathematical analysis,Convex set,Subderivative,Convex function,Regularization (mathematics),Rate of convergence,Local convergence,Iterated function,Mathematics,Bounded function | Journal |
Volume | Issue | ISSN |
46 | 5 | 0363-0129 |
Citations | PageRank | References |
7 | 0.65 | 4 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| William W. Hager | 1 | 1603 | 214.67 |
| Hong-Chao Zhang | 2 | 32 | 4.37 |